Bloch-Zener Oscillations across a Merging Transition of Dirac Points

Lih-King Lim, Jean-Noel Fuchs, and Gilles Montambaux

Laboratoire de Physique des Solides, CNRS UMR 8502, Universite Paris-Sud, F-91405 Orsay cedex, France(Received 12 January 2012; published 26 April 2012)

Bloch oscillations are a powerful tool to investigate spectra with Dirac points. By varying band

parameters, Dirac points can be manipulated and merged at a topological transition toward a gapped

phase. Under a constant force, a Fermi sea initially in the lower band performs Bloch oscillations and may

Zener tunnel to the upper band mostly at the location of the Dirac points. The tunneling probability is

computed from the low-energy universal Hamiltonian describing the vicinity of the merging. The

agreement with a recent experiment on cold atoms in an optical lattice is very good.

DOI: 10.1103/PhysRevLett.108.175303 PACS numbers: 67.85.Lm, 03.75.Lm, 37.10.Jk, 73.22.Pr

Introduction.—Dirac points in energy bands occur inspecial two-dimensional (2D) condensed matter systems[1], such as graphene [2], nodal points in d-wave super-conductors, and surface states of three-dimensional (3D)topological insulators [3]. They are fascinating instances ofultrarelativistic behavior emerging as low-energy effectivedescription of electrons in solids. Dirac points are bandtouching points that carry a topological charge, namely, aBerry phase ��. In most systems, Dirac points occur indipolar pairs (the so-called fermion doubling). Undervariation of external parameters, it is possible to movethese Dirac points and even make them merge. This merg-ing signals a topological (Lifshitz) transition between agapless phase with a disconnected Fermi surface to agapped phase [4–7]. For example, a uniaxial stress ingraphene leads to a motion of the Dirac points but themerging transition is not reachable [8]. The quasi-2Dorganic conductor �-ðBEDT-TTFÞI32 is a good candidateto observe this transition under pressure [9]; however, ithas not been realized yet.

Recently, a new type of experiment, realized with ultra-cold atoms loaded in a 2D optical lattice, has provided analternative way to study Dirac points [10]. By combiningtechniques of Bloch oscillations and adiabatic mapping ofcold atoms, the band structure of the system can be studiedwith momentum resolution [11–14]. Specifically, the ex-periment of ETH Zurich [10] utilizes such techniques for anoninteracting Fermi gas in a tunable two-band systemfeaturing Dirac points. Their existence is revealed throughLandau-Zener (LZ) tunneling from the lower to the upperband. As the lattice amplitude is varied, a drastic change inthe transferred atomic fraction provides a qualitative sig-nature of the Dirac points and their merging.

In this Letter, we present a complete description ofLandau-Zener tunneling through a pair of Dirac points,using a universal low-energy Hamiltonian describing themerging transition [7]. We show how the transferred frac-tion provides a key signature of the merging transition andthat it depends crucially on the direction of the motion withrespect to the merging direction. We find a very good

agreement between the computed averaged LZ probabil-ities and the experimental data. Furthermore, new experi-mental signatures for varying Bloch oscillations and acoherent Stuckelberg interferometry are presented.Tight-binding model.—We consider a nearest-neighbor

tight-binding model on a square lattice. The four hoppingamplitudes between neighbors are taken as t, t along y- andt0, t00 along x-direction (see Fig. 1). When t0 � t00, there aretwo inequivalent sites—called A and B—per unit cellgiving rise to two bands. When t00 ¼ 0, a link is brokenrealizing a brick-wall lattice, which has the same connec-tivity as the honeycomb lattice albeit with a rectangulargeometry. When t0 ¼ t00, it is a standard square lattice withanisotropic amplitudes along x and y and a single site perunit cell. The Hamiltonian (with nearest-neighbor distancea � 1 and @ � 1) reads

H ¼ 0 fðkÞf�ðkÞ 0

� �(1)

with fðkÞ ¼ �ðteiky þ te�iky þ t0eikx þ t00e�ikxÞwhere thehopping amplitudes are positive and k ¼ ðkx; kyÞ is the

Bloch wave vector. The energy spectrum is given by�ðkÞ ¼ �jfðkÞj. It features two Dirac cones when t0 þt00 < 2t (gapless D phase) and a gap when t0 þ t00 > 2t

FIG. 1 (color online). (a) Square lattice indicating the hoppingamplitudes and the two inequivalent sites. (b) Band structure inthe gapless D phase for t0 ¼ t ¼ 0:2, t00 ¼ 0:05 in units of ER,see text. The first Brillouin zone is indicated by the square.

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(gapped G phase). At t0 þ t00 ¼ 2t the Dirac points mergeat k ¼ ð0; �Þ and there is a single touching point betweenthe two bands. When t0 ¼ t00, the band structure is that of asquare lattice, with lines of Dirac points (L phase), whichbecomes isotropic when t0 ¼ t00 ¼ t (I phase). This modelcan therefore describe the transition between the G and Dphase, and moreover, the crossover from the D to L phase.In the following, energies are measured in units of therecoil energy ER ¼ �2

@2=ð2ma2Þ where m is the atomic

mass.Mapping to the universal Hamiltonian.—In a crystal

which is time-reversal and inversion symmetric, mergingcan only occur atG=2 points whereG is a reciprocal latticevector. Near such a point, it is possible to write a minimallow-energy Hamiltonian that captures the topological tran-sition and describes both Dirac cones at once [7]. Close tothe merging, an expansion for small q ¼ k�G=2 givesrise to an effective Hamiltonian which has the universalform

H ¼ 0 �� þ q2y2m� � icxqx

�� þ q2y2m� þ icxqx 0

0@

1A (2)

and a spectrum � ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið q2y2m� þ��Þ2 þ c2xq

2x

q. The model

depends on three independent parameters ��, m�, and cx.The merging transition is driven by the parameter ��hereafter called the merging gap. When �� < 0, the spec-

trum contains two Dirac points at qD ¼ ð0;� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�j��j

p Þand j��j represents the energy of the saddle points con-necting them, which is located at q ¼ 0 (D phase). Whenincreasing �� toward 0, the two Dirac points approacheach other along the qy direction until they merge when

�� ¼ 0. Exactly at the merging, there is a single touchingpoint between the two bands, with a semi-Dirac (or hybrid)

spectrum � ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðq2y=2m�Þ2 þ ðcxqxÞ2

q[5,15]. By increas-

ing the driving parameter still further, a true gap of magni-tude 2�� > 0 opens at q ¼ 0 (G phase). We map thetight-binding to the universal model by comparing theirenergy expansions near the Dirac points [7]. In the Dphase, we find �� ¼ t0 þ t00 � 2t and the Dirac cone ve-

locities are cx ¼ t0 � t00 and cy ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4t2 � ðt0 þ t00Þ2p

, so that

the mass is obtained from m� ¼ �2��=c2y ¼ 2=ð2tþ t0 þt00Þ. In the G phase, �� and cx are unchanged andm� ¼ 1=ð2tÞ.

Landau-Zener tunneling with Dirac cones.—Consideratoms initially in the lower band performing Bloch oscil-lations under the influence of a constant applied force F.By accelerating these atoms in the vicinity of a Dirac point,their tunneling probability to the upper band is finite, aproblem considered by Landau and Zener [16]. In thefollowing, the universal low-energy Hamiltonian is usedto compute the interband transition probability.

Motion along the kx direction: Single Dirac cone.—Inthe D phase, an atom moving along the kx direction

encounters at most one Dirac cone during a single Blochoscillation [Fig. 2(b)]. The LZ probability for such a linearavoided band crossing is given by [16]

PxZ ¼ e��ðgap=2Þ2=cxF ¼ e��ðq2y=2m�þ��Þ2=cxF; (3)

where qy is the position with respect to the merging

point G=2 ¼ ð0; �Þ. Note that PxZ ¼ 1 for qy ¼ �qD ¼

� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�2m���p

, positions of the two Dirac points. ActuallyEq. (3) is not only valid in the D but extends to the Gphase across the merging transition. This quantity is shownin Fig. 2(a) as a function of the transverse momentum qyand ��.As the experiment is performed with a cloud of non-

interacting fermions, we need also to average the LZprobability over the initial distribution of atoms. We con-sider a 2D cloud of harmonically trapped fermions at zerotemperature for a filling fraction sufficiently smaller thanhalf-filling. The energy spectrum close to k ¼ 0 is �ðkÞ �k2x=ð2mxÞ þ k2y=ð2myÞ (as measured from the band bottom)

with the band masses mx ¼ ð2tþ t0 þ t00Þ=½4t0t00 þ 2tðt0 þt00Þ� and my ¼ 1=ð2tÞ. The semiclassical energy of an atom

is therefore �ðk; rÞ ¼ k2x=ð2mxÞ þ k2y=ð2myÞ þ ðmx!xx2 þ

my!2yy

2Þ=2 where !x=2� and !y=2� are the trapping

frequencies [17]. The fraction �x of atoms transferred tothe upper band is then given by the averaged probability�x ¼ hPx

Zi where

FIG. 2 (color online). Motion along kx. (a) LZ probability PxZ

[Eq. (3)] as a function of ���=F and of the transverse momen-tum qy. Here cx=F � 0:5 and m�F � 0:13. (b) Trajectories

along the kx direction. (c) Transferred fraction to the upperband �x as a function of ���=F for different sizes kFy of the

cloud.

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h� � �i ¼R�ðk;rÞ<�F

dkxdkydxdy � � �R�ðk;rÞ<�F

dkxdkydxdy(4)

and �F ¼ k2Fx=ð2mxÞ ¼ k2Fy=ð2myÞ is the Fermi energy,

which defines kFx and kFy [18]. The transferred fraction

�x as a function of �� and of the size kFy of the cloud is

shown in Fig. 2(c). For a cloud of finite size kFy, only a

finite proportion of atoms may tunnel to the upper bandwhen �� < 0.

Motion along the ky direction: Double Dirac cone.—In

theG phase, the tunneling probability is vanishingly small.In the following, we concentrate on the D phase whereatoms performing one Bloch oscillation in the ky direction

have the possibility to encounter two inequivalent Diraccones successively. The scenario is therefore richer thanbefore since the tunneling process implies two successiveLandau-Zener events [see Fig. 3(b)]. The probability Py

Z

associated with each LZ event is now

PyZ � e�2�� ¼ e��ðc2xq2x=cyFÞ ¼ e��ðc2xq2x=F

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2j��j=m�p

Þ; (5)

which defines the adiabaticity parameter �. In the follow-ing, we calculate the total interband probability Py

t asso-ciated with the two successive events, in the limit wherethey can be considered independent. Quantitatively, the LZ

tunneling time �maxð ffiffiffiffi�

p; �Þ=cxqx [19] should be shorter

than the time 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m�j��j

p=F it takes an atom to travel

between the two Dirac points, i.e., not too close to themerging transition.First assuming that the two tunneling events are inco-

herent, we combine the probabilities to find the interbandtransition probability [18,19]

Pyt ¼ 2Py

Zð1� PyZÞ; (6)

which is shown in Fig. 3(a) as a function of cx and thetransverse momentum qx. Notice that Py

t vanishes whenqx ¼ 0 because Py

Z ¼ 1. For an initial cloud of size kFx, thetransferred fraction is �y ¼ hPy

t i where the average is

defined in Eq. (4) [18]. The result is shown in Fig. 3(c).Comparison to the experiment.—The ETH Zurich ex-

periment is performed on a harmonically trapped 3D Fermigas loaded in a 2D optical trap [10]. Our model treats atrapped 2D Fermi gas in a 2D band structure. The opticallattice potential is Vðx; yÞ ¼ �V �Xcos

2ð�xþ �=2Þ �VXcos

2ð�xÞ � VYcos2ð�yÞ � 2�

ffiffiffiffiffiffiffiffiffiffiffiffiVXVY

pcosð�xÞ cosð�yÞ,

where� ¼ 0:9, � ¼ �, and the laser wavelength is 2awithamplitudes VY ¼ 1:8, 0 V �X 6:5, and 0 VX 1. Tomake a precise comparison, we perform single-particlenumerical band structure calculation provided by the 2Doptical potential using a truncated plane-wave expansion,and establish a map between the optical lattice parametersand that of the universal Hamiltonian of Eq. (2) for�� < 0.The latter is done by fitting the parameters of the universalHamiltonian to the two lowest lying energy bands in thevicinity of the Dirac cones [18]. The corresponding (t, t0,t00)-tight-binding model is then obtained by the analyticalmapping [see paragraph after Eq. (2)]—the main featuresthat are not captured are the Dirac cones tilting and theparticle-hole asymmetry, which appear not to be relevant.Qualitatively, t stays roughly constant in the consideredrange of optical lattice parameters. The hopping t0 in-creases when VX � V �X increases, whereas t00 increaseswhen VX þ V �X decreases. Finally, we compute the trans-ferred fractions �x and �y as a function of VX and V �X,

shown in Fig. 4. For the experimental parametersF ¼ 0:02, kFy ’ �=2, and kFx ’ 2 (corresponding to �F �0:4), we find a striking agreement with Figs. 4a and 4b ofRef. [10].Discussion.—First consider the case of the motion along

kx. The line of maximum transfer probability �x (redregion in Fig. 4(a)] corresponds to a maximal LZ proba-bility Px

Z � 1 for a large number of atoms. Playing with theaveraging order gives �x � exp½��hðq2y=2m� þ��Þ2i=ðcxFÞ�, in which hq2yi ¼ k2Fy=6 and hq4yi ¼ k4Fy=16.

Its maximum occurs when �� ’ �hq2yi=ð2m�Þ, which ex-

plains why it is near the merging line �� ¼ 0, but slightlyinside theD phase, as seen experimentally. The transferredfraction �x is a symmetric function of its natural variable��—when doing the average properly it is actually slightlyasymmetric—and its width reduces when decreasing cx(by decreasing VX). Both features are seen experimentally,

FIG. 3 (color online). Motion along ky. (a) Total probabilityPyt for atoms tunneling to the upper band [Eq. (6)] as a function

of cx=F and of the transverse momentum qx. Here ��=F ¼ �5and m�F � 0:13. (b) Double LZ tunneling along the ky direc-

tion. (c) Transferred fraction �y as a function of cx=F for various

sizes kFx of the initial cloud (kFx¼ 1; �=2; 2).

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see Fig. 4a of Ref. [10]. Obviously, the width and themaximum should increase when increasing F.

Adding a staggered on-site potential ��=2 opens a gap� in the spectrum. Experimentally, this gap is controlledby the parameter � of the potential Vðx; yÞ. It was foundthat the decay of the transferred fraction �x as a function of� is well fitted by a Gaussian (see Fig. 2b of Ref. [10]).Here, we prove that it is of the form �xð�Þ ¼ �xð0Þexpð���2=ð4cxFÞÞ and that � � 4:3ð�=�� 1Þ for theexperimental parameters V �X;X;Y ¼ ½3:6; 0:28; 1:8�.

Next, consider the case of the motion along ky. The

interband transition probability [Eq. (6)] is a nonmono-tonic function of the LZ probability Py

Z and it is maximumwhen Py

Z ¼ 1=2. This explains the existence of the

maximum of �y well inside the D phase (red region in

Fig. 4(b)]. When playing with the averaging order, �y �e�Xð1� e�XÞ=2 where X � �c2xhq2xi=cyF. The maximum

occurs when e�X � 1=2, i.e., when c2x=cy ’F ln2=ð�hq2xiÞ ¼ const. Compared to the previous case,�y is a much more asymmetric function of its natural

variable X / c2x=cy: the decay at large X is slower than at

small X (the signal extends more toward large cx, which isseen experimentally). In contrast to the previous case, theposition of the line of maxima depends on F, as shown inFigs. 4(b) and 4(c), but its amplitude is almost independent.Its width decreases when cx ! 0. Furthermore, �y vanishes

as VX ! 0, eventually reaching the square lattice limit(L phase).Since the two LZ events along ky are expected to be

coherent, Stuckelberg interferences in the transferred frac-tion �y should be observable. Equation (6) should indeed

be replaced by Pyt ¼ 4Py

Zð1� PyZÞcos2ð’=2þ ’dÞ where

’ ¼ 4RqD0 dqy

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�� þ q2y=2m

�Þ2 þ c2xq2x

q=F is a dynami-

cally acquired phase in between the two tunneling eventsand ’d ¼ ��=4þ �ðln�� 1Þ þ arg�ð1� i�Þ is a phasedelay given in terms of the gamma function and � isdefined in Eq. (5) [18,19]. Averaging Py

t over the 2Datomic distribution gives the transferred fraction �y shown

in Fig. 4(d). However, such interferences are not observedin the ETH experiment, which we attribute to averagingover the third spatial direction. Briefly, interference fringesin Fig. 4(d) are lines of constant �� with a fringe spacing�0:04ER. For the experimentally given trapping frequencyin the z direction, we estimate that �� varies along z by�0:03ER. This should be enough to wash out theinterferences.Conclusion.—Landau-Zener tunneling conveniently

probes the energy spectrum in the vicinity of Dirac points.Depending on the direction of the applied force, the atomsexperience a LZ transition through a single or a pair ofDirac cones. We calculated the transferred fraction in theframework of the universal Hamiltonian describing themerging transition, and found a very good agreementwith the ETH experiment. To summarize, the importantparameters are the merging gap �� and the velocity cxperpendicular to the merging direction. A simplified phasediagram is shown in Fig. 4(e). Although the transferthrough a single Dirac cone probes the merging transition,the transfer through a pair of cones signals a doubleLandau-Zener event inside the D phase and a crossovertoward the L phase.As perspectives, we expect Stuckelberg interferences to

be observable in the strictly 2D regime. Furthermore, itshould now be possible to tune the optical lattice right atthe merging transition and to study the semi-Dirac spec-trum, thus opening the way to explore new phenomena. Forexample, applying an artificial Uð1Þ gauge potential[20,21] should reveal unusual Landau levels [5].

D

L

G

I

PZx 1

PZy 1/2

-

c x

0

ee

FIG. 4 (color online). (a) Transferred fraction �x as a functionof the optical lattice parameters V �X and VX (here kFy ¼ �=2,

F ¼ 0:02). Inset: lines of constant ��. (b) Transferred fraction�y (kFx ¼ 2, F ¼ 0:02). Inset: lines of constant c2x=cy. (c) Same

as (b) with F ¼ 0:1. (d) Same parameters as (b) taking coherenceinto account and leading to Stuckelberg oscillations. (e) Phasediagram in the (� ��, cx) plane showing the G, D, L, and Iphases (see text). Px

Z ¼ 1 along the merging transition (continu-

ous line). The crossover line Pyz ¼ 1=2 corresponds to cx /

F1=2j��j1=4 and is plotted for two different forces (dashed anddotted lines). The color code for (a)–(d) is such that �m ¼ 0:5,0.3, 0.3, and 0.6, respectively. The black line in (a)–(d) indicatesthe merging transition �� ¼ 0.

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We acknowledge support from the Nanosim Grapheneproject under Grant No. ANR-09-NANO-016-01

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